HSC Extension 1 and 2 Mathematics/3-Unit/Preliminary/Harder 2-Unit

Implicit differentiation

Implicit differentiation is a method of differentiating an expression in $x$ and $y$, where $x$ and $y$ are related in some manner and neither are constant.

For example, one could differentiate $f(x)=y^{2}$ with respect to $x$ as follows:

Using the chain rule:

${\begin{aligned}{\tfrac {df}{dx}}&={\tfrac {df}{dy}}\times {\tfrac {dy}{dx}}\\&=2y\times {\tfrac {dy}{dx}}\end{aligned}}$

It is useful to think of implicit differentiation as normal differentiation with respect to $x$, only whenever you come across a term with $y$, you multiply the differentiated term by $dy/dx$.

Another example: find the derivative $dy/dx$ of $x^{2}+y^{2}=r^{2}$

Working:

${\begin{aligned}2x+2y.{\frac {dy}{dx}}&=0\\2x&=-2y.{\frac {dy}{dx}}\\{\frac {dy}{dx}}&=-{\frac {x}{y}}\end{aligned}}$